1. Numerical analysis of a locking-free mixed
fem for a bending moment formulation of
Reissner-Mindlin plates
¸ ˜
L OURENC O B EIR A O DA V EIGA1 , DAVID M ORA2 , R ODOLFO R ODR´GUEZ3
I
1
`
Dipartimento di Matematica, Universita degli Studi di Milano.
2
´
Departamento de Matematica, Universidad del B´o B´o.
ı ı
3
´ ´
Departamento de Ingenier´a Matematica, Universidad de Concepcion.
ı
´ ´ ´
Sesion Especial de Analisis Numerico
´
XX Congreso de Matematica Capricornio COMCA 2010
´
Universidad de Tarapaca, Arica, Chile
4–6 de Agosto de 2010.

2. Contents
• The model problem
• The continuous formulation
• Galerkin formulation
• Numerical tests
Mixed FEM for Reissner-Mindlin plates. –2– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

3. The model problem.
Given g ∈ L2 (Ω), find β , γ and w
− div(C(ε(β))) − γ = 0 in Ω,
−div γ = g in Ω,
κ
γ = 2 (∇w − β) in Ω,
t
w = 0, β = 0 on ∂Ω,
• w is the transverse displacement,
• β = (β1 , β2 ) are the rotations,
• γ is the shear stress,
• t is the thickness,
• κ := Ek/2(1 + ν) is the shear modulus (k is a correction factor),
E
• Cτ := 12(1−ν 2 ) ((1 − ν)τ + ν tr(τ )I) ,
• we restrict our analysis to convex plates.
Mixed FEM for Reissner-Mindlin plates. –3– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

4. The model problem. (cont.)
We introduce as a new unknown the bending moment σ = (σij )1≤i,j≤2 , defined by
σ := C(ε(β)).
12(1 − ν 2 ) 1 ν
• C −1 τ := τ− 2)
tr(τ )I .
E (1 − ν) (1 − ν
We rewrite the equation above as:
−1 1
C σ = ∇β − (∇β − (∇β)t ) = ∇β − rJ,
2
1
• r := − 2 rot β ,
 
0 1
• J :=  .
−1 0
Mixed FEM for Reissner-Mindlin plates. –4– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

5. The continuous formulation.
Continuous mixed problem
Find ((σ, γ), (β, r, w)) ∈ H × Q such that
t2
C −1 σ : τ + γ·ξ+ β · (div τ + ξ) + r(τ12 − τ21 ) + wdiv ξ = 0
Ω κ Ω Ω Ω Ω
η · (div σ + γ) + s(σ12 − σ21 ) + vdiv γ = − gv,
Ω Ω Ω Ω
for all ((τ , ξ), (η, s, v)) ∈ H × Q, where
H := H(div; Ω) × H(div ; Ω),
Q := [L2 (Ω)]2 × L2 (Ω) × L2 (Ω),
with
H(div; Ω) := {τ ∈ [L2 (Ω)]2×2 : div τ ∈ [L2 (Ω)]2 },
and
H(div ; Ω) := {ξ ∈ [L2 (Ω)]2 : div ξ ∈ L2 (Ω)}.
Mixed FEM for Reissner-Mindlin plates. –5– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

6. The continuous formulation. (cont.)
Continuous mixed problem
a((σ, γ), (τ , ξ)) + b((τ , ξ), (β, r, w)) = 0 ∀(τ , ξ) ∈ H,
b((σ, γ), (η, s, v)) = − gv ∀(η, s, v) ∈ Q,
Ω
where
t2
a((σ, γ), (τ , ξ)) := C −1 σ : τ + γ · ξ,
Ω κ Ω
b((τ , ξ), (η, s, v)) := η · (div τ + ξ) + s(τ12 − τ21 ) + vdiv ξ.
Ω Ω Ω
In the analysis we will utilize the following t-dependent norm for the space H
(τ , ξ) H := τ 0,Ω + div τ + ξ 0,Ω +t ξ 0,Ω + div ξ 0,Ω ,
while for the space Q, we will use
(η, s, v) Q := η 0,Ω + s 0,Ω + v 0,Ω .
Mixed FEM for Reissner-Mindlin plates. –6– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

7. The continuous formulation. (cont.)
Additional regularity
Proposition 1 Suppose that Ω is a convex polygon and g ∈ L2 (Ω). Then, there
a
exists a constant C , independent of t and g , such that
w 2,Ω + β 2,Ω + γ H(div ;Ω) +t γ 1,Ω + σ 1,Ω +t div σ 1,Ω + r 1,Ω ≤C g 0,Ω .
a
D. N. A RNOLD AND R. S. FALK, A uniformly accurate finite element method for the Reissner-Mindlin plate,
SIAM J. Numer. Anal., 26 (1989) 1276–1290.
Mixed FEM for Reissner-Mindlin plates. –7– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

8. The continuous formulation. (cont.)
Ellipticity in the kernel
V := {(τ , ξ) ∈ H : ξ + div τ = 0, τ = τ t and div ξ = 0 in Ω}.
Lemma 1 There exists C > 0, independent of t, such that
2
a((τ , ξ), (τ , ξ)) ≥ C (τ , ξ) H ∀(τ , ξ) ∈ V.
Proof. Given (τ , ξ) ∈ V , using tr(τ )2 ≤ 2(τ : τ ) ∀τ ∈ [L2 (Ω)]2×2 , we obtain
12(1 − ν) 2 t2 2
a((τ , ξ), (τ , ξ)) ≥ τ 0,Ω + ξ 0,Ω .
E κ
Since div τ + ξ 0,Ω = 0 and div ξ 0,Ω = 0, we get
2 2
a((τ , ξ), (τ , ξ)) ≥ C τ 0,Ω + div τ + ξ 0,Ω + t2 ξ 2
0,Ω + div ξ 2
0,Ω ,
2
⇒ a((τ , ξ), (τ , ξ)) ≥ C (τ , ξ) H,
Mixed FEM for Reissner-Mindlin plates. –8– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

9. The continuous formulation. (cont.)
Inf-Sup condition
Lemma 2 There exists C > 0, independent of t, such that
|b((τ , ξ), (η, s, v))|
sup ≥ C (η, s, v) Q ∀(η, s, v) ∈ Q.
(τ ,ξ)∈H (τ , ξ) H
(τ ,ξ)=0
Well-posedness
Theorem 2 There exists a unique ((σ, γ), (β, r, w)) ∈ H × Q solution of the
continuous mixed problem and
((σ, γ), (β, r, w)) H×Q ≤C g 0,Ω ,
where C is independent of t.
Mixed FEM for Reissner-Mindlin plates. –9– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

10. Galerkin formulation.
¯
• {Th }h>0 : regular family of triangulations of the polygonal region Ω.
• hT : diameter of the triangle T ∈ Th .
• h := max{hT : T ∈ Th }.
¯
• Ω= {T : T ∈ Th }.
Finite elements subspaces
γ
Hh := {ξh ∈ H(div ; Ω) : ξh |T ∈ RT0 (T ), ∀T ∈ Th },
Qw := {vh ∈ L2 (Ω) : vh |T ∈ P0 (T ), ∀T ∈ Th },
h
Mixed FEM for Reissner-Mindlin plates. – 10 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

11. Galerkin formulation. (cont.)
We consider the unique polynomial bT ∈ P3 (T ) that vanishes on ∂T and is
normalized by T bT = 1.
B(Th ) := {τ ∈ H(div ; Ω) : (τi1 , τi2 )|T ∈ span{curl(bT )}, i = 1, 2, ∀T ∈ Th } ,
where curl v := (∂2 v, −∂1 v).
Hh := {τ h ∈ H(div ; Ω) : τ h |T ∈ [RT0 (T )t ]2 , ∀T ∈ Th } ⊕ B(Th ),
σ
Qβ := {ηh ∈ [L2 (Ω)]2 : ηh |T ∈ [P0 (T )]2 , ∀T ∈ Th },
h
Qr := sh ∈ H 1 (Ω) : sh |T ∈ P1 (T ), ∀T ∈ Th ,
h
Note that Hh
σ
× Qβ × Qr correspond to the PEERSa finite elements.
h h
a
D. N. A RNOLD, F. B REZZI AND J. D OUGLAS, PEERS: A new mixed finite element for the plane elasticity,
Japan J. Appl. Math., 1 (1984) 347–367.
Mixed FEM for Reissner-Mindlin plates. – 11 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

12. Galerkin formulation. (cont.)
Discrete mixed problem:
Find ((σ h , γh ), (βh , rh , wh )) ∈ Hh × Qh such that
a((σ h , γh ), (τ h , ξh )) + b((τ h , ξh ), (βh , rh , wh )) = 0 ∀(τ h , ξh ) ∈ Hh ,
b((σ h , γh ), (ηh , sh , vh )) = − gvh ∀(ηh , sh , vh ) ∈ Qh .
Ω
γ
Hh := Hh × Hh ,
σ
Qh := Qβ × Qr × Qw .
h h h
Mixed FEM for Reissner-Mindlin plates. – 12 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

13. Galerkin formulation. (cont.)
Discrete kernel
Vh := (τ h , ξh ) ∈ Hh : ηh · (div τ h + ξh ) + sh (τ12h − τ21h )
Ω Ω
+ vh div ξh = 0 ∀(ηh , sh , vh ) ∈ Qh .
Ω
Let (τ h , ξh )∈ Vh . Taking (0, 0, vh ) ∈ Qh and using that (div ξh )|T is a constant,
since vh |T is also a constant, we conclude that div ξh = 0 in Ω.
Now, taking (ηh , 0, 0)∈ Qh , since div τ h = 0 in Ω ∀τ h ∈ B(Th ), we have that
(div τ h )|T is a constant vector. Since div ξh = 0, we have that ξh |T is also a
constant vector. Therefore, since ηh |T is also a constant vector, we conclude that
(div τ h + ξh ) = 0 in Ω. Thus, we obtain
Vh = (τ h , ξh ) ∈ Hh : ξh + div τ h = 0, sh (τ12h − τ21h ) = 0 ∀sh ∈ Qr
h
Ω
and div ξh = 0 in Ω .
Mixed FEM for Reissner-Mindlin plates. – 13 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

14. Galerkin formulation. (cont.)
Ellipticity in the discrete kernel
Lemma 3 There exists C > 0 such that
2
a((τ h , ξh ), (τ h , ξh )) ≥ C (τ h , ξh ) H ∀(τ h , ξh ) ∈ Vh ,
where the constant C is independent of h and t.
Discrete inf-Sup condition
Lemma 4 There exists C > 0, independent of h and t, such that
|b((τ h , ξh ), (ηh , sh , vh ))|
sup ≥ C (ηh , sh , vh ) Q ∀(ηh , sh , vh ) ∈ Qh .
(τ h ,ξh )∈Hh (τ h , ξh ) H
(τ h ,ξh )=0
Mixed FEM for Reissner-Mindlin plates. – 14 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

15. Galerkin formulation. (cont.)
Theorem 3 There exists a unique ((σ h , γh ), (βh , rh , wh ))
∈ Hh × Qh solution of
the discrete mixed problem. Moreover, there exist C, C > 0, independent of h and t,
such that
((σ h , γh ), (βh , rh , wh )) H×Q ≤C g 0,Ω ,
and
((σ, γ), (β, r, w)) − ((σ h , γh ), (βh ,rh , wh )) H×Q
≤C inf ((σ, γ),(β, r, w)) − ((τ h , ξh ), (ηh , sh , vh )) H×Q ,
(τ h ,ξh )∈Hh
(ηh ,sh ,vh )∈Qh
where ((σ, γ), (β, r, w)) ∈ H × Q is the unique solution of the continuous mixed
problem.
Mixed FEM for Reissner-Mindlin plates. – 15 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

16. Galerkin formulation. (cont.)
Rate of convergence
Theorem 4 Let ((σ, γ), (β, r, w)) ∈ H × Q and
((σ h , γh ), (βh , rh , wh )) ∈ Hh × Qh be the unique solutions of the continuous and
discrete mixed problem, respectively. If g ∈ H 1 (Ω), then,
((σ, γ), (β, r, w)) − ((σ h , γh ), (βh , rh , wh )) H×Q ≤ Ch g 1,Ω .
Mixed FEM for Reissner-Mindlin plates. – 16 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

17. Numerical tests.
• Isotropic and homogeneous plate.
• Ω := (0, 1) × (0, 1).
• t = 0.001.
• E = 1, ν = 0.30 and k = 5/6.
Figure 1: Square plate: uniform meshes.
Mixed FEM for Reissner-Mindlin plates. – 17 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

18. Numerical tests. (cont.)
Choosing the load g as:
E
g(x, y) = 12y(y − 1)(5x2 − 5x + 1) 2y 2 (y − 1)2
12(1 − ν 2 )
+x(x − 1)(5y 2 − 5y + 1) + 12x(x − 1)(5y 2 − 5y + 1) 2x2 (x − 1)2
+y(y − 1)(5x2 − 5x + 1) ,
so that
1 3
w(x, y) = x (x − 1)3 y 3 (y − 1)3
3
2t2
− y 3 (y − 1)3 x(x − 1)(5x2 − 5x + 1)
5(1 − ν)
+ x3 (x − 1)3 y(y − 1)(5y 2 − 5y + 1) ,
β1 (x, y) =y 3 (y − 1)3 x2 (x − 1)2 (2x − 1),
β2 (x, y) =x3 (x − 1)3 y 2 (y − 1)2 (2y − 1).
Mixed FEM for Reissner-Mindlin plates. – 18 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

19. Numerical tests. (cont.)
e(σ) := σ − σ h 0,Ω , e(γ) := γ − γh t,H(div ;Ω) ,
e(β) := β − βh 0,Ω , e(r) := r − rh 0,Ω , e(w) := w − wh 0,Ω ,
log(e(·)/e′ (·))
rc(·) := −2 ,
log(N/N ′ )
where N and N ′ denote the degrees of freedom of two consecutive triangulations with
errors e and e′ .
Mixed FEM for Reissner-Mindlin plates. – 19 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

20. Numerical tests. (cont.)
Table 1: Errors and experimental rates of convergence for variables σ and γ , computed
on uniform meshes.
N e(σ) rc(σ) e(γ) rc(γ)
1345 0.40270e-04 – 0.31715e-02 –
5249 0.19649e-04 1.054 0.15876e-02 1.016
20737 0.09760e-04 1.019 0.07942e-02 1.008
82433 0.04868e-04 1.008 0.03971e-02 1.004
328705 0.02431e-04 1.004 0.01986e-02 1.002
Mixed FEM for Reissner-Mindlin plates. – 20 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

21. Numerical tests. (cont.)
Table 2: Errors and experimental rates of convergence for variables β , r and w, com-
puted on uniform meshes.
N e(β) rc(β) e(r) rc(r) e(w) rc(w)
1345 0.39713e-04 – 0.87462e-04 – 0.66226e-05 –
5249 0.18189e-04 1.147 0.39217e-04 1.178 0.27707e-05 1.280
20737 0.08884e-04 1.043 0.15009e-04 1.398 0.13136e-05 1.086
82433 0.04416e-04 1.013 0.05491e-04 1.457 0.06478e-05 1.025
328705 0.02205e-04 1.004 0.01991e-04 1.466 0.03228e-05 1.007
Mixed FEM for Reissner-Mindlin plates. – 21 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

22. Numerical tests. (cont.)
10
σ ♦
1 γ +
β
0.1 ⋆ r ×
⋆ w △
⋆
0.01 ⋆ h⋆ ⋆
+
0.001 +
+
e +
+
0.0001 ×
♦ ×
♦ ×
1e-05 △ ♦ ×
△ ♦
△ ♦
×
1e-06 △
△
1e-07
1e-08
1000 10000 100000 1e+06
degrees of freedom N
Mixed FEM for Reissner-Mindlin plates. – 22 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

23. Numerical tests. (cont.)
Figure 2: Approximate transverse displacement (left) and first component of the rotation
vector (right).
Mixed FEM for Reissner-Mindlin plates. – 23 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

24. Numerical tests. (cont.)
Figure 3: Approximate shear vector: first component (left) and second component
(right).
Mixed FEM for Reissner-Mindlin plates. – 24 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

25. Numerical tests. (cont.)
Figure 4: Approximate bending moment: σ 11h (left) and σ 12h (right).
Mixed FEM for Reissner-Mindlin plates. – 25 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

26. Numerical tests. (cont.)
Figure 5: Approximate bending moment: σ 21h (left) and σ 22h (right).
Mixed FEM for Reissner-Mindlin plates. – 26 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I

27. Many thanks for your attention.
Mixed FEM for Reissner-Mindlin plates. – 27 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I